On the other hand, a Cantor set consists of infinitely many pieces --- in fact, uncountably many pieces. Moreover, each piece is a point and every point in the Cantor set is a limit point of other points in this set. So a Cantor set should be visualized as a "cloud" of points---no two points touching, but infinitely many points scattered in any region around a given point.
So the fundamental dichotomy says that Julia sets for x2 + c come in one of two varieties: connected sets (one piece) or Cantor sets (infinitely many pieces). There is no in-between: there are no c-values for which Jc consists of 10 or 20 or 756 pieces.
How do we decide what shape a given Jc assumes? Amazingly, it is the orbit of 0 that determines this. For if the orbit of 0 tends to infinity under iteration of x2 + c, then the fact is that Jc is a Cantor set. On the other hand, if the orbit of 0 does not tend to infinity, then Jc is a connected set.
A visual way to view this dichotomy is given by the Mandelbrot set.
If c lies in
M, then we know that the orbit of 0 does
not escape to infinity under iteration of x2 + c, so
Jc
must be connected. If c does not lie in M then Jc
is a Cantor set. This dichotomy thus gives us a second interpretation of
the Mandelbrot set.
The Mandelbrot set consists of all c-values for which
It is amazing that the orbit of 0 "knows" the shape of the Julia
set for x2 + c. The reason that 0 is so special
stems from the fact that 0 is the critical point of x2
+ c. That is, the derivative of x2 + c is 2x,
and this derivative only vanishes at x=0.